RIMS Kôkyûroku Bessatsu B32 (2012), Sage (Sage for number theorists) By (Iwao Kimura) Abstract Sage is an open source software for computer al

Size: px

Start display at page:

Download "RIMS Kôkyûroku Bessatsu B32 (2012), Sage (Sage for number theorists) By (Iwao Kimura) Abstract Sage is an open source software for computer al"

えいしろうあくや
4 years ago
Views:

1 RIMS Kôkyûroku Bessatsu B32 (2012), Sage (Sage for number theorists) By (Iwao Kimura) Abstract Sage is an open source software for computer algebra and numerical computation. The aim of the Sage project is to create a viable free open source alternative to Magma, Maple, Mathematica and Matlab. In this article, we give a brief introduction to Sage for number theorists. 1. Sage Sage free Sage Magma, Maple, Mathematica, Matlab W. Stein University of Washington Sage 1.1. Sage The Sage Notebook Sage Sage Sage 1 Received April 10, Revised February 28, Mathematics Subject Classification(s): 11Y40 Key Words: Sage, Computational number theory iwao@sci.u-toyama.ac.jp c 2012 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

126 1. Sagenb Linux KNOPPIX/Math 7 KNOPPIX/Math DVD Microsoft Windows Linux Sage KNOPPIX/Math 29 1.2. Sage Microsoft Windows, Apple MacOS X, Linux Sage Sage

2 Sagenb Linux KNOPPIX/Math 7 KNOPPIX/Math DVD Microsoft Windows Linux Sage KNOPPIX/Math Sage Microsoft Windows, Apple MacOS X, Linux Sage Sage Battery included 1 OS Apple MacOS X, Linux Ubuntu OS Microsoft Windows Windows Linux VMWare player 26, VirtualBox 13 Sage Sage OS 400MB 1

3 Sage (Sage for number theorists) CUI Sage./sage Character User Interface Sage Sage quit 1.4. GUI Sage GUI Graphical User Interface Sage Sage GUI GUI GUI Google Chrome, Mozilla Firefox, Internet Explorer, Apple Safari TEX Sage notebook() Sage 1. Sage : $. / sage 4.6/ sage Sage Version 4. 6, Release Date : Type notebook ( ) f o r the GUI, and l i c e n s e ( ) f o r i n f o r m a t i o n. sage : notebook ( ) New Worksheet evaluate Shift+Enter Shift+ TinyMCE 2 Javascript TinyMCE $ L A TEX Sage Sage Sage

4 128 Sage notebook? GUI Sage Save & quit Sign out 1.5. Sage Sage Magma, Maple, Mathematica Matlab Sage Sage Python Sage Ginac 1, Singular 5, Maxima 9 Pari-gp 14 mwrank 4 Sage Sage Sage Sage Pari-gp Sage Sage GPL v2 2 BSD, Apache License, MIT License 3 Sage Sage Sage 2 GNU General Public License, Versin 2, 3

5 Sage (Sage for number theorists) 129 Sage 5 2. Python Sage Python Python Guid van Rossum Python 4 Linuix, Apple MacOS X, Microsoft Windows Python Python Unicode Python Sage Python Python C {, } C Python CUI 1.3 Python Python Python Tutorial 25 if sage: else...: # Sage Sage Python 2.6 Python Sage Python 3

6 130 sage : a = 1 # a s s i g n 1 to a v a r i a b l e a. sage : i f a==1:.... : print """a is 1""".... : else :.... : print """a is not 1""" a i s 1 sage : 3. sage : i f a==1:.... : print """a is 1""" # missing t a b causes an e r r o r. I n d e n t a t i o n E r r o r : expected an indented block (<ipython console >, l i n e 2) Python for [] Python 0 4. for sage : a = [ one, two, three ] sage : for x in a :.... : print x, l e n ( x ) one 3 two 3 t h r e e 5 sage : print a [ 1 ] two : {, } print x,, 5. for sage : d={ one : 1, two : 2, three : 3 } sage : d. keys ( ) # t h e l i s t o f keys [ three, two, one ] sage : d. v a l u e s ( ) # t h e l i s t o f v a l u e s [ 3, 2, 1 ] sage : for k in d. i t e r k e y s ( ) :.... : print d [ k ], 3 2 1

7 Sage (Sage for number theorists) 131 d.iterkeys() range() range() sage : for x in range ( 1 0 ) : print x, # note t h a t t r a i l i n g,.... : sage : for x in range ( 1, 1 0, 3 ) : print x,.... : list comprehension sum() 8. sage : [ xˆ2 for x in range ( 1 0 ) ] # square o f 0, 1,..., 9. [ 0, 1, 4, 9, 16, 25, 36, 49, 64, 8 1 ] sage : sum ( [ xˆ2 for x in range ( 1 0 ) ] ) 285 double x sage : def double ( x ) :.... : return 2 x.... : sage : double ( 1 0 ) Sage Python Python 2 x**2 3. Sage Sage

8 Fermat sage : f a c t o r (2ˆ2ˆ5+1) # prime f a c t o r i z a t i o n o f t h e 5 th Fermat number Q QQ sage : R.<x> = PolynomialRing (QQ) ; R U n i v a r i a t e Polynomial Ring in x over Rational F i e l d PolynomialRing() 6 R.<x>=QQ[]??? sage : f = 2 xˆ2+3 x+1; f 2 xˆ2 + 3 x + 1 sage : f a c t o r? # sage : f a c t o r ( f ) # polynomial f a c t o r i z a t i o n ( 2 ) ( x + 1/2) ( x + 1) sage : f. f a c t o r ( ) # t h i s i s the same as above. f 7 7 Q sage : Q7 = Qp( 7 ) ; Q7 # the f i e l d o f p a d i c numbers, here p = 7 7 a d i c F i e l d with capped r e l a t i v e p r e c i s i o n 20 sage : S.<x>=Q7 [ ] ; S U n i v a r i a t e Polynomial Ring in x over 7 a d i c F i e l d with capped r e l a t i v e p r e c i s i o n 20 sage : f 7=S ( f ) ; f 7 # c o n v e r s i o n from QQ[ x ] to Q7 [ x ] (2 + O(7ˆ20) ) xˆ2 + (3 + O(7ˆ20) ) x + (1 + O(7ˆ20) ) sage : f a c t o r ( f 7 ) 6 CamelCase 7 PolynomialRing f factor

9 Sage (Sage for number theorists) 133 (2 + O(7ˆ20) ) ( ( 1 + O(7ˆ20) ) x + (1 + O(7ˆ20) ) ) ( ( 1 + O(7ˆ20) ) x + ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ19 + O( 7 ˆ 2 0) ) ) 3 Q / p p F b = 2 b 3 (mod p) p sage : p = next prime ( 1 0 ˆ 5 ) ; p # the prime next to 10ˆ sage : F = F i n i t e F i e l d ( p ) ; F # the f i n i t e f i e l d o f p elements F i n i t e F i e l d o f s i z e sage : b=f. m u l t i p l i c a t i v e g e n e r a t o r ( ) ; b # a p r i m i t i v e r o o t mod p 2 sage : F ( 3 ). l o g ( b ) # the d i s c r e t e l o g o f 3 w. r. t. b sage : FF.<a> = F i n i t e F i e l d ( p ˆ 2 ) ; FF F i n i t e F i e l d in a o f s i z e ˆ2 a p Sage Stein 18 2 d 2 kk F kk.gen(), F.gen() sage : kk.<d>=quadraticfield ( 974) ; kk Number F i e l d i n d with d e f i n i n g polynomial xˆ sage : kk. class number ( ) 36 sage : C=kk. c l a s s g r o u p ( ) ; C Class group o f order 36 with s t r u c t u r e C12 x C3 o f Number F i e l d i n d with d e f i n i n g polynomial xˆ sage : C. e l e m e n t a r y d i v i s o r s ( ) [ 3, 1 2 ] sage : F.<d>=QuadraticField (199) ; F # a r e a l q u a d r a t i c f i e l d Number F i e l d i n d with d e f i n i n g polynomial xˆ2 199 sage : F. class number ( ) # i t s c l a s s number 1

10 134 sage : F. u n i t s ( ) # i t s u n i t group ( non fundamental?) [ d ] sage : e = F. u n i t s ( ) [ 0 ] ; e d sage : 1/e # fundamental d sage : gp. quadunit (4 199) w 2 Pari-gp quadunit() sage : k=cyclotomicfield ( 7 ) ; k Cyclotomic F i e l d o f order 7 and degree 6 sage : p r i n t k. class number ( ), k. u n i t s ( ) 1 [ zeta7 ˆ5 + zeta7, zeta7 ˆ4 + zeta7 ˆ3 + 1 ] sage : k. s u b f i e l d s ( ) # Galois 16. ) sage : G=k. g a l o i s g r o u p ( ) ; G G a l o i s group o f Cyclotomic F i e l d o f order 7 and degree 6 sage : p2=k. prime above ( 2 ) ; p2 F r a c t i o n a l i d e a l ( zeta7 ˆ5 zeta7 ˆ3 zeta7 ˆ2) sage : Z=p2. decomposition group ( ) ; Z Subgroup [ ( ), ( 1, 3, 4 ) ( 2, 5, 6 ), ( 1, 4, 3 ) ( 2, 6, 5 ) ] o f G a l o i s group o f Cyclotomic F i e l d o f order 7 and degree 6 sage : Z. f i x e d f i e l d ( ) (Number F i e l d i n zeta70 with d e f i n i n g polynomial xˆ2 + x + 2, Ring morphism : From : Number F i e l d i n zeta70 with d e f i n i n g polynomial xˆ2 + x + 2 To : Cyclotomic F i e l d o f order 7 and degree 6 Defn : zeta70 > zeta7 ˆ4 + zeta7 ˆ2 + zeta7 ) sage : s=p2. a r t i n s y m b o l ( ) ; s ( 1, 3, 4 ) ( 2, 5, 6 ) sage : s ( k. gen ( ) ) # k. gen ( ) i s the g e n e r a t o r o f k zeta7 ˆ2 Sage Sage

11 Sage (Sage for number theorists) 135 proof=false 17. proof=false sage : CyclotomicField ( 2 3 ). class number ( p r o o f=false ) 3 23 CUI Control+C GUI Action Interrupt 18. sage : from sage. misc. c i t a t i o n import g e t s y s t e m s sage : g e t s y s t e m s ( k. subfields () ) [ PARI, MPFI, FLINT, MPFR, GMP, NTL ] 15 Pari-gp 14, MPFI 8, Flint 11, MPFR 10, GMP 6, NTL sage : E = E l l i p t i c C u r v e ([ 82, 0 ] ) ; E # i f only 2 arg s a, b are given, then i t means yˆ2 = x ˆ3 + ax + b. E l l i p t i c Curve d e f i n e d by yˆ2 = xˆ3 82 x over Rational F i e l d sage : E. d i s c r i m i n a n t ( ). f a c t o r ( ) # good red. o u t s i d e 2, 41. 2ˆ9 41ˆ3 sage : E. gens ( ) # g e n e r a t o r o f r a t i o n a l p o i n t s. [( 9 : 3 : 1), ( 8 : 12 : 1), ( 1 : 9 : 1) ] sage : E. has cm ( ) # t h i s i s a CM e l l i p t i c curve. True sage : L=E. l s e r i e s ( ) ; L Complex L s e r i e s o f the E l l i p t i c Curve d e f i n e d by yˆ2 = xˆ3 82 x over Rational F i e l d sage : L. t a y l o r s e r i e s ( ) ( e 22) z + ( e 22) z ˆ z ˆ z ˆ z ˆ5 + O( z ˆ6) sage : E. a n a l y t i c r a n k ( ) 3 L 1,

12 136 p p h p h p = wq χ ( B ) 1,χ, 2 w p 1 2p, Q Hasse 1 p Dirichlet χ B 1,χ χ 1 Bernoulli Washington 27, Chap. 4, Th Sage Dirichlet DirichletGroup() Dirichlet Dirichlet Bernoulli Gauss Jacobi 20. Dirichlet sage : DG=DirichletGroup ( 2 3 ) ; sage : c h i=dg. gen ( ) # a g e n e r a t o r sage : c h i D i r i c h l e t c h a r a c t e r modulo 23 o f conductor 23 mapping 5 > zeta22 sage : c h i. b e r n o u l l i ( 1 ) # t h e 1 s t B e r n o u l l i number 6/23 zeta22 ˆ9 + 14/23 zeta22 ˆ8 + 6/23 zeta22 ˆ7 2/23 zeta22 ˆ6 + 12/23 zeta22 ˆ5 10/23 zeta22 ˆ4 8/23 zeta22 ˆ3 14/23 zeta22 ˆ2 18/23 zeta22 16/23 sage : c h i. gauss sum ( ) # Dirichlet Stein 17, Chap. 4 p p Bernoulli B 1,χ /2 8 return 21. p sage : def hpminus analytic ( p ) :.... : DG = DirichletGroup ( p ).... : c h i = DG. gens ( ) [ 0 ].... : return 2 p prod ([ (( c h i ) ˆ(2 k+1) ). b e r n o u l l i ( 1 ) /2 for k in range ( 0, ( p 1) /2) ] ).... : sage : hpminus analytic ( 2 3 ) 3 n hnminus analytic() r""" """ hnminus analytic?

13 Sage (Sage for number theorists) n relativeclassno.sage def hnminus analytic ( n ) : r """ This function computes the relative class number of an n-th cyclotomic field by the analytic class number formula. """ w = CyclotomicField ( n ). z e t a o r d e r ( ) i f i s p r i m e p o w e r ( n ) : Q = 1 else : Q=2 DG = DirichletGroup ( n ) return w Q prod ( ( c h i. b e r n o u l l i ( 1 ) ) /2 for c h i in DG i f c h i. i s o d d ( ) ) Sage relativeclassno.sage 22 Python Sage attach() Sage load() 23. Sage sage : attach "~/ Lang/Sage/relativeclassno.sage" Magma 28 p h p 24 2 ZZ() Sage hpminus analytic() conversion sage : pandhpm=[(p, hpminus analytic ( p ) ) for p in primes ( 2 0, ) ] ; sage : pandhpmandfactor =[( t [ 0 ], t [ 1 ], f a c t o r (ZZ( t [ 1 ] ) ) ) for t in pandhpm ] Sage Sage Sage

14 138 sage : sage : db save ( pandhpmandfactor, pandhpmandfactor ) # save db ( pandhpmandfactor ) # l o a d Sage J. Cremona mwrank 4 mwrank -b mwrank sage : E2012=E l l i p t i c C u r v e ( [ ˆ 2, 0 ] ) ; E2012 E l l i p t i c Curve d e f i n e d by yˆ2 = xˆ x over Rational F i e l d sage : Emw=E2012. mwrank( o p t i o n s= -b 15 ) ; Emw # y 2 = x x 2012 Sage Q( d) ε d 2 E d : y 2 = x ε d 26. Q( 41) sage : Q41.<a>=QuadraticField ( 4 1 ) ; sage : eps = UnitGroup (Q41). fundamental units ( ) [ 0 ] ; sage : E = E l l i p t i c C u r v e (Q41, [ 0, 1728 eps ] ) ; E E l l i p t i c Curve d e f i n e d by yˆ2 = xˆ3 + (8640 a+55296) over Number F i e l d in a with d e f i n i n g polynomial xˆ2 41 sage : dscnt = E. simon two descent ( verbose =1) ; dscnt # sage : E( dscnt [ 2 ] [ 0 ] ) ( /93025 a /93025 : / a / : 1) sage : E. rank ( ) 2 sage : E. gens ( ) [( /93025 a /93025 : / a / : 1) ] sage : E(Q41) Abelian group o f p o i n t s on E l l i p t i c Curve d e f i n e d by yˆ2 = xˆ3 + (8640 a+55296) over Number F i e l d in a with d e f i n i n g polynomial xˆ2 41 d = d =

15 Sage (Sage for number theorists) 139 ell.gp 16 ell.gp Pari-gp D. Simon Simon Sage Sage sagesupport 5 Sage OS, 3.3. Sage 27. sage : o p t i o n a l p a c k a g e s ( ) # sage : i n s t a l l p a c k a g e ( d a t a b a s e c r e m o n a e l l c u r v e ) J. Cremona 3 Cremona 11 a Cremona sage : c=cremonadatabase ( ) ; c Cremona database o f e l l i p t i c curves sage : E11=c. a l l c u r v e s ( 1 1 ) ; E11 { a1 : [ [ 0, 1, 1, 10, 20], 0, 5 ], a3 : [ [ 0, 1, 1, 0, 0 ], 0, 5 ], a2 : [ [ 0, 1, 1, 7820, ], 0, 1 ] } Cremona J. Jones 6 Odlyzko Riemann zeta Sloane OEIS 4. Parent/Element, Category, Coercion and Conversion Sage Parent/Element Sage Euclid

16 Categories of ZZ sage : ZZ. c a t e g o r i e s ( ) [ Category o f e u c l i d e a n domains, Category o f p r i n c i p a l i d e a l domains, Category o f gcd domains, Category o f i n t e g r a l domains, Category o f commutative r i n g s, Category o f domains, Category o f r i n g s, Category o f rngs, Category o f commutative a d d i t i v e groups, Category o f semirings, Category o f commutative a d d i t i v e monoids, Category o f commutative a d d i t i v e semigroups, Category o f a d d i t i v e magmas, Category o f monoids, Category o f semigroups, Category o f magmas, Category o f s e t s, Category o f s e t s with p a r t i a l maps, Category o f o b j e c t s ] sage : QQ. c a t e g o r i e s ( ) # 1 1/2 3/2 1 1/2 1 + (1/2) 1 1/2 Sage 1 1/ Category of 1, 1/2 sage : 1. c ategory ( ) Category o f elements o f I n t e g e r Ring sage : 1. c ategory ( )==ZZ # c aution! False sage : (1/2). c a t e g o r y ( ) Category o f elements o f Rational F i e l d sage : (1+1/2). c a t e g o r y ( ) Category o f elements o f Rational F i e l d coerce Sage 1 1 Parent 1 Element Parent/Element Magma Stein 20 Sage Category Parent/Element 31. Parent/Element

17 Sage (Sage for number theorists) 141 sage : 1. parent ( ) I n t e g e r Ring sage : ( 1 / 2 ). parent ( ) Rational F i e l d sage : (1+1/2). parent ( ) Rational F i e l d x 1 x 1/2 x 1 Sage 32. x + (1/2) sage : R.<x> = ZZ [ ] ; sage : x. parent ( ) U n i v a r i a t e Polynomial Ring in x over I n t e g e r Ring sage : ( 1 / 2 ). parent ( ) Rational F i e l d sage : ( x +1/2). parent ( ) U n i v a r i a t e Polynomial Ring in x over Rational F i e l d Sage x 1/2 Parent R sage : R. c o n s t r u c t i o n ( ) ( Poly [ x ], I n t e g e r Ring ) sage : QQ. c o n s t r u c t i o n ( ) ( F r a c t i o n F i e l d, I n t e g e r Ring ) 33. Parent Z Integer Ring R = Z[x] 1 coercion 1 coercion 34. coercion sage : cm=sage. s t r u c t u r e. element. g e t c o e r c i o n m o d e l ( ) sage : cm. d i s c o v e r c o e r c i o n (R, QQ) ( Conversion map : From : U n i v a r i a t e Polynomial Ring i n x over I n t e g e r Ring To : U n i v a r i a t e Polynomial Ring i n x over Rational Field, Polynomial base i n j e c t i o n morphism : From : Rational F i e l d To : U n i v a r i a t e Polynomial Ring i n x over Rational F i e l d ) x + (1/2) Z[x] Q(x) Q[x] Sage

18 142 Coercion, Parent/Element Sage reference manual 22 The Coercion Model Sage Sage sage : ( 1 0 / 2 ). parent ( ) Rational F i e l d sage : 10/2==ZZ(10/2) True /2 Parent 5 conversion 12, 132 conversion convert 5. Sage Sage Sage org/help.html#sagestandarddoc Sage tutorial 23 Stein 21 Kosan Sage W. Stein 19 Sage Sage Sage sage-support sage-devel sage-japan URL asksage Sage Magma, Maple, Mathematica Matlab free Sage Sage

19 Sage (Sage for number theorists) Ginac is not a CAS, 2 Moxiecode Systems AB., TinyMCE, 3 J. E. Cremona, Elliptic Curve Data, University of Warwick, uk/ masgaj/ftp/data/. 4, mwrank and related programs for elliptic curves over Q, University of Warwick, masgaj/mwrank/. 5 G.-M.; Pfister G.; Schönemann H. Decker, W.; Greuel, Singular A computer algebra system for polynomial computations, (2010), 6 The GMP developers, GMP, the GNU Multiple Precision arithmetic library, edition 4.3.2, FSF, Jan 2011, 7 The KNOPPIX/Math developing team, KNOPPIX/Math 2011 Japanese edition, knoppix v6.4.4-math-dvd ja.iso, Fabrice Rouillier et. al., MPFI, multiple precision interval packages, INRIA, 2010, https: //gforge.inria.fr/projects/mpfi. 9 The Maxima Group, Maxima, a Computer Algebra System. Version , maxima.sourceforge.net/. 10 Guillaume Hanrot, Vincent Lefévre, Patrick Pélissier, Philippe Théveny, and Paul Zimmermann, MPFR, multiple precision floating-point reliable library, version , FSF, June 2010, 11 William Hart, Fredrik Johanssony, and Sebastian Pancratzz, FLINT version 2.1.0, 9 March 2011, 12 Ted Kosan, SAGE for newbies, Feb. 2008, tkosan/newbies book/sage for newbies v1.23.pdf. 13 Oracle, VirtualBox, Oracle, 14 PARI Group, Bordeaux, PARI/GP, Version 2.4.3, 2008, available from 15 V. Shoup, NTL, a library for doing number theory, August 2009, net/ntl/. 16 D. Simon, ell.gp, Universié Caen, March 2011, simon/ ell.gp. 17 William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007, With an appendix by Paul E. Gunnells. MR (2008d:11037) 18, Elementary number theory: primes, congruences, and secrets, a computational approach, Undergraduate Texts in Mathematics, Springer, New York, MR (2009i:11002) 19, Mathematical software and me: A very personal recollection, Dec 2009, http: //wstein.org/mathsoftbio/history.pdf.

20 144 20, Brief history and motivation behind the Sage coercion model, Tech. report, Nov. 2010, blog post on brief-history-and-motivation-behind.htm. 21, Three lectures about explicit methods in number theory using Sage, Release 4.6.2, March 2011, /. 22 Sage Development Team, Sage, Reference Manual, Release 4.6.2, March 2011, http: // 23 The Sage Development Team, Sage, Tutorial, Release 4.6.2, March 2011, sagemath.org/doc/tutorial/. 24 Ted Kosan, SAGE, , ponpoko/knoppix/sage for newbies ja.pdf. 25 Guido van Rossum and Fred L. Drake, JR., Python tutorial release 2.7.1, Python Software Foundation, March , available from 26 VMWare, Inc., VMWare player, VMWare, Inc., player/overview.html. 27 Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Springer-Verlag, New York, MR 97h: , Magma, 7 (,,,, and, eds.), 2009, pp ,, 49 (2010), no. 9, 8 14.

2.2 Sage I 11 factor Sage Sage exit quit 1 sage : exit 2 Exiting Sage ( CPU time 0m0.06s, Wall time 2m8.71 s). 2.2 Sage Python Sage 1. Sage.sage 2. sa

2.2 Sage I 11 factor Sage Sage exit quit 1 sage : exit 2 Exiting Sage ( CPU time 0m0.06s, Wall time 2m8.71 s). 2.2 Sage Python Sage 1. Sage.sage 2. sa I 2017 11 1 SageMath SageMath( Sage ) Sage Python Sage Python Sage Maxima Maxima Sage Sage Sage Linux, Mac, Windows *1 2 Sage Sage 4 1. ( sage CUI) 2. Sage ( sage.sage ) 3. Sage ( notebook() ) 4. Sage